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Internal division

Coordinates of the point C which divides the line segment joining the points A(x1, y1) and B(x2, y2) internally in the ratio m : n are given by
 
x = 70021.png

External division

Coordinates of the point C which divides the line segment joining the points A(x1, y1) and B(x2, y2) externally in the ratio m : n are given by
 
x = 70013.png
 
Notes:
  • The midpoint of (x1y1) and (x2y2) is 70529.png.
  • The centroid (G) (concurrency point of medians of a triangle) of a triangle having vertices (x1y1), (x2y2), and (x3,y3) is G 70523.png using the fact that centroid is concurrency point of medians, and G divides median AD in ratio 2:1.
  • Orthocenter of a triangle is point of concurrency of altitudes of triangle.
  • In a triangle (not equilateral) orthocenter (H) (concurrency point of altitudes of triangle), centroid (G), and circumcenter (O) are collinear and in the ratio HG/GO= 2.
  • If circumcenter of a triangle (concurrency point of perpendicular bisector of sides of triangle) with vertices (x1y1), (x2y2), and (x3y3) is origin (0, 0) then orthocenter is (x1 + x2 + x3y1 + y2 +y3).
  • Incenter of a triangle (concurrency point of internal angle bisector) with verticesA(x1y1), B(x2y2), and C(x3y3) is I70517.png using thefact that internal angle bisector ADdivides BC in the ratio AB/AC.
Excenters
 
Let A(x1y1), B(x2y2), and C(x3y3) be the vertices of the triangle ABC, and let ab, andc be the lengths of the sides BCCA, and ABrespectively.
70990.png

The circle which touches the sides BC and two sides AB and AC produced is called the escribed circle opposite to the angle A.

The bisectors of the external angle B andC meet at a point I1 which is the center of the escribed circle opposite to the angle A.

The coordinates of I1 are given by
70984.png
 
The coordinates of I2 and I3 (centers of escribed circles opposite to the angles B and C respectively) are given by
 
70978.png
and 70972.png
  • To prove that ABCD are vertices of
Parallelogram Show that diagonals AC and BD bisect each other.
Rhombus Show that diagonals AC and BD bisect each other and adjacent sides AB and BC are equal.
Square Show that diagonals AC and BD equal and bisect each other and adjacent sides AB and BC are equal.
Rectangle Show that diagonals AC and BD equal and bisect each other.





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